Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$, with $s$ in quadrant $I$, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.
A) $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$

B) $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$

C) $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$

D) $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$

Question

Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$, with $s$ in quadrant $I$, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.
A) $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$

B) $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$

C) $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$

D) $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$

Quizplus · Accepted Answer

The sum formula for cosine is $\cos(s + t) = \cos s \cos t - \sin s \sin t$. Given $\cos s = \frac{1}{3}$ and $\sin t = \frac{1}{4}$, we find $\sin s$ and $\cos t$ using Pythagoras' theorem. For $s$ in quadrant I, $\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(\frac{1}{3}\right)^2} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}$. For $t$ in quadrant II, where $\cos$ is negative, $\cos t = -\sqrt{1 - \sin^2 t} = -\sqrt{1 - \left(\frac{1}{4}\right)^2} = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4}$. Substituting these into the sum formula gives $\cos(s + t) = \left(\frac{1}{3}\right)\left(-\frac{\sqrt{15}}{4}\right) - \left(\frac{2\sqrt{2}}{3}\right)\left(\frac{1}{4}\right) = -\frac{\sqrt{15}}{12} - \frac{2\sqrt{2}}{12} = -\frac{\sqrt{15} + 2\sqrt{2}}{12}$.

Find the Exact Value of the Expression Using the Provided cos⁡(s+t)\cos ( s + t )cos(s+t)

Find the Exact Value of the Expression Using the Provided $\cos ( s + t )$